On a four-parameter generalization of some special sequences

Silva, Robson da; Oliveira, Kelvin Souza de; Neto, Almir Cunha da Graça

doi:10.1016/j.dam.2018.03.002

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…Corollary 3. For r = a and s = b, the connection formula in (11) reduces to the following connection formula:…”

Section: Connection Formulae Between the Two Generalized Classes Of F...mentioning

confidence: 99%

“…If we set r = 2 and s = −1 in Relation (10), then after making use of the orthogonality relation of T j (x), Relation ( 28) can be obtained. Relation ( 29) can be obtained by setting a = 2 and b = −1 in Relation (11) and making use of the orthogonality relation of U j (x).…”

Section: Corollarymentioning

confidence: 99%

“…A type of generalized Fibonacci polynomials, namely, distance Fibonacci polynomials, was investigated in [10]. A four-parameter generalization of some special sequences was also considered in [11]. Some other studies regarding different types of generalized sequences can be found for example in [12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

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Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals

2022

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The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind 2F1(z) are included in all connection coefficients for a specific z. Several new connection formulae between some famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, and Fermat–Lucas polynomials, are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed.

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“…Corollary 3. For r = a and s = b, the connection formula in (11) reduces to the following connection formula:…”

Section: Connection Formulae Between the Two Generalized Classes Of F...mentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

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Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals

2022

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“…Many mathematicians, working primarily in the field of combinatorics and discrete mathematics, began to deal with the other sequences which are a similar generalization of the Fibonacci sequence (see [34,[37][38][39][40][41][42][43][44][45]). In [34], the authors noted that from recurrence relation (13) of the sequence (F 2 (q, n)) n≥0 immediately follows that:…”

Section: Binet-like Form Of General Linear Recurrencementioning

confidence: 99%

On Coding by (2,q)-Distance Fibonacci Numbers

Matoušová

Trojovský

2020

Mathematics

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In 2006, A. Stakhov introduced a new coding/decoding process based on generating matrices of the Fibonacci p-numbers, which he called the Fibonacci coding/decoding method. Stakhov’s papers have motivated many other scientists to seek certain generalizations by introducing new additional coefficients into recurrence of Fibonacci p-numbers. In 2013, I. Włoch et al. studied (2,q)-distance Fibonacci numbers F2(q,n) and found some of their combinatorial properties. In this paper, we state a new coding theory based on the sequence (Tq(n))n=−∞∞, which is an extension of Włoch’s sequence (F2(q,n))n=0∞.

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“…In the last years, many papers were devoted to the study of some special sequences or special elements, especially in their connections with particular cases of algebras obtained by the Cayley-Dickson (quaternions, octonions). Some of them are: [Ba, Pr; 09], [Ca;16], [Ci,Ip;16], [Fa,Pl;07(1)], [Fa,Pl;07(2)], [Fl;09], [Fl,Sa;15(1)], [Fl,St;09], [Ha,15], [Mo;17], [Si,18].…”

Section: Introductionmentioning

confidence: 99%